In this thesis, we study the topological constraints on the signature and the Euler characteristic $(sigma(X), e(X))$ for smooth $4$-manifolds $X$ (or complex surfaces $X$) which are surface bundles over surfaces with nonzero signature. The first main result is about the improved upper bounds for the minimal base genus function $b(f,n)$ for a fixed fiber genus $f$ and a fixed signature $4n$. In particular, we construct new smooth $4$-manifolds with a fixed signature $4$ and small Euler characteristic which are surface bundles over surfaces by subraction of Lefschetz fibrations. They include an example with the smallest Euler characteristic among known examples with non-zero signature. Secondly, we explore possibilities to construct Kodaira fibrations with small signature which are smooth surface bundles over surfaces as ramified coverings of products of two complex curves. To obtain the minimal base genus and the smallest possible signature, we investigate the action of the monodromy of the fibration of pointed curves. Throughout the paper we`ll see that the surface mapping class group plays an important role in both constructions and the control of topological invariants.